4 Problem Sheet 4:
Computations with Galois groups
Exercise 4.1 Let \(K\subset M\subset L\) be a tower of finite extensions.
\(\;(a)\) Given that \(L/K\) is a Galois extension, prove that \(L/M\) is as well.
\(\;(b)\) Give an example where \(L/K\) is Galois but \(M/K\) is not.
Exercise 4.2 Which of the following simple extensions \(L/K\) are Galois? Give reasons.
\(\;(a)\;\) \(L=\mathbb{Q}(\sqrt[3]{2})\) over \(K=\mathbb{Q},\hspace{5em}\) \(\;(b)\;\) \(L=\mathbb{Q}(\sqrt[4]{2})\) over \(K=\mathbb{Q},\)
\(\;(c)\;\) \(L=\mathbb{Q}(\sqrt[8]{2})\) over \(K=\mathbb{Q}(\sqrt{2}),\hspace{2.85em}\) \(\;(d)\;\) \(L=K(\sqrt[8]{2})\) over \(K=\mathbb{Q}(i),\)
\(\;(e)\;\) \(L=\mathbb{F}_5(x)\) over \(K=\mathbb{F}_5(x^4),\hspace{3.45em}\) \(\;(f)\;\) \(L=\mathbb{F}_p(x)\) over \(K=\mathbb{F}_p(x^2)\) for \(p\) prime,
\(\;(g)\;\) \(L=\mathbb{C}(x)\) over \(K=\mathbb{C}(x^5),\hspace{4.05em}\) \(\;(h)\;\) \(L=\mathbb{F}_2(x)\) over \(K=\mathbb{F}_2(x^4).\)
Exercise 4.3 \(\;(a)\) Find a minimal Galois extension \(L/\mathbb{Q}\) containing \(\mathbb{Q}(\sqrt[4]{5})\).
\(\;(b)\) Describe the structure of \(G=\Gal(L/\mathbb{Q})\).
\(\;(c)\) Find all the proper subgroups of \(G\). Which ones are normal subgroups?
\(\;(d)\) Find all intermediate fields \(\mathbb{Q}\subset M\subset L\) with \([M:\mathbb{Q}]=2\) or \(4\). Which are normal extensions of \(\mathbb{Q}\)?
Exercise 4.4 Show that \(\sqrt{7}\not\in L=\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})\) by considering how automorphisms in the Galois group \(\Gal(L/\mathbb{Q})\) act on \(\sqrt{7}\).
Exercise 4.5 \(\;(a)\) Prove that \(L=\mathbb{Q}\left(\sqrt{-3},\sqrt[3]{2},\sqrt[3]{5}\right)\) is Galois over \(\mathbb{Q}\).
\(\;(b)\) Let \(K=\mathbb{Q}(\sqrt{-3})\). Show that \([L:K]=9\).
\(\;(c)\) Find the structure of the group \(H=\Gal\left(L/K\right)\).
\(\;(d)\) Find all intermediate fields \(K\subset M\subset L\) such that \([L:M]=3\).
\(\;(e)\) Find the structure of the group \(G=\Gal(L/\mathbb{Q})\).
Exercise 4.6 In each of the following cases, for the field \(K\) and polynomial \(f(x)\in K[x]\), find the splitting field \(L\) of \(f(x)\) over \(K\). Also find the Galois group \(G=\Gal(L/K)\) and all the proper subgroups \(H\subset G\).
\(\;(a)\;\;K=\mathbb{Q}\) and \(f(x)=(x^2-2)(x^2-5),\hspace{1em}\) \(\;(b)\;\;K=\mathbb{Q}\) and \(f(x)=(x^4+1)(x^2-5),\)
\(\;(c)\;\;K=\mathbb{Q}\) and \(f(x)=x^5-1,\hspace{5.2em}\) \(\;(d)\;\;K=\mathbb{Q}(i)\) and \(f(x)=x^8-2,\)
\(\;(e)(\star)\;\;K=\mathbb{Q}\) and \(f(x)=x^6-2,\hspace{3.9em}\) \(\;(f)\;\;K=\mathbb{Q}(\sqrt{-3})\) and \(f(x)=(x^3-2)(x^3-5),\)
\(\;(g)(\star)\;\;K=\mathbb{Q}\) and \(f(x)=x^4+x^2-1,\hspace{1.7em}\) \(\;(h)\;\;K=\mathbb{Q}(i)\) and \(f(x)=x^8+1.\)
Exercise 4.7 Let \(K\) be a field with \(\ch(K)\neq 2\).
\(\;(a)\) Given \(a,b\in K\), show that \([K(\sqrt{a},\sqrt{b}):K]=4\) if and only if \(a,b,ab\not\in{K^\times}^2\).
\(\;(b)\) We say a set \(\{a_1,...,a_n\}\subset K^\times\) is independent modulo squares when \[m_i\in\mathbb{Z}\;\;\text{and}\;\; a_1^{m_1}a_2^{m_2}\cdots a_n^{m_n}\in {K^\times}^2\;\;\implies\;\;\text{all $m_i$ are even}.\] \(\;\;\;\;\,\) Let \(L=K(\sqrt{a_1},\sqrt{a_2},...,\sqrt{a_n})\) for some \(a_i\in K\). Prove by induction that \([L:K]=2^n\) if and only if \(\{a_1,...,a_n\}\) is independent modulo squares.
\(\;(c)\) In that case, show that \(L/K\) is Galois with \(\Gal(L/K)=\underbrace{\mathbb{Z}/2\times\cdots\times\mathbb{Z}/2}_\text{$n$ times}\).
Exercise 4.8 Show that the following extensions \(L/K\) are Galois. Furthermore, find the Galois group \(G=\Gal(L/K)\), the subgroups \(H\subset G\) and the corresponding fixed fields \(L^H\).
\(\;(a)\;\) \(L=\mathbb{R}(x)\) over \(K=\mathbb{R}(x^2+x^{-2}),\)
\(\;(b)\;\) \(L=\mathbb{R}(x,y)\) over \(K=\mathbb{R}(x^2,y^2),\)
\(\;(c)\;\) \(L=\mathbb{R}(x,y)\) over \(K=\mathbb{R}(x^2+y^2,xy).\)
Exercise 4.9 Let \(G\) be the group of automorphisms of the field \(L=\mathbb{F}_3(x)\) given by \[\sigma_{a,b}:x\mapsto ax+b\quad\text{where $a,b\in\mathbb{F}_3$ and $a\neq 0$.}\] Find all subgroups of \(G\) and subfields \(M\) with \(L^G\subset M\subset L\).